Find the value of k if the following equations have equal roots
i) `x^(2)–2(1+3k)x+7(3+2k)= 0`
ii) `x^(2) -15 - k(2x – 8) = 0`
iii) `(3k+1)x^(2)+2(k+1)x+k=0`
iv) `x^(2)+2(k+2)x+9k=0`

Answer

To find the value of \( k \) for the given equations to have equal roots, we will use the condition that the discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) must be equal to zero. The discriminant is given by:

\[ D = b^2 - 4ac \]

We will analyze each equation one by one.

### 1. First Equation:
\[ x^2 - 2(1 + 3k)x + 7(3 + 2k) = 0 \]

Here,
- \( a = 1 \)
- \( b = -2(1 + 3k) \)
- \( c = 7(3 + 2k) \)

Now, we calculate the discriminant \( D \):

\[ D = \left(-2(1 + 3k)\right)^2 - 4 \cdot 1 \cdot 7(3 + 2k) \]

Calculating \( D \):

\[ D = 4(1 + 3k)^2 - 28(3 + 2k) \]

Setting \( D = 0 \):

\[ 4(1 + 6k + 9k^2) - 84 - 56k = 0 \]

Simplifying:

\[ 36k^2 - 8k - 80 = 0 \]

Dividing by 4:

\[ 9k^2 - 2k - 20 = 0 \]

Factoring:

\[ (9k + 10)(k - 2) = 0 \]

Thus, \( k = -\frac{10}{9} \) or \( k = 2 \).

### 2. Second Equation:
\[ x^2 - 15 - k(2x - 8) = 0 \]

Rearranging gives:

\[ x^2 - 2kx + (8k - 15) = 0 \]

Here,
- \( a = 1 \)
- \( b = -2k \)
- \( c = 8k - 15 \)

Calculating the discriminant \( D \):

\[ D = (-2k)^2 - 4(1)(8k - 15) \]

Setting \( D = 0 \):

\[ 4k^2 - 32k + 60 = 0 \]

Dividing by 4:

\[ k^2 - 8k + 15 = 0 \]

Factoring:

\[ (k - 5)(k - 3) = 0 \]

Thus, \( k = 3 \) or \( k = 5 \).

### 3. Third Equation:
\[ (3k + 1)x^2 + (2(k + 1))x + k = 0 \]

Here,
- \( a = 3k + 1 \)
- \( b = 2(k + 1) \)
- \( c = k \)

Calculating the discriminant \( D \):

\[ D = [2(k + 1)]^2 - 4(3k + 1)(k) \]

Setting \( D = 0 \):

\[ 4(k^2 + 2k + 1) - 4(3k^2 + k) = 0 \]

Simplifying:

\[ 4k^2 + 8k + 4 - 12k^2 - 4k = 0 \]

Combining like terms:

\[ -8k^2 + 4k + 4 = 0 \]

Dividing by -4:

\[ 2k^2 - k - 1 = 0 \]

Factoring:

\[ (2k + 1)(k - 1) = 0 \]

Thus, \( k = -\frac{1}{2} \) or \( k = 1 \).

### 4. Fourth Equation:
\[ x^2 + 2(k + 2)x + 9k = 0 \]

Here,
- \( a = 1 \)
- \( b = 2(k + 2) \)
- \( c = 9k \)

Calculating the discriminant \( D \):

\[ D = [2(k + 2)]^2 - 4(1)(9k) \]

Setting \( D = 0 \):

\[ 4(k^2 + 4k + 4) - 36k = 0 \]

Simplifying:

\[ 4k^2 + 16k + 16 - 36k = 0 \]

Combining like terms:

\[ 4k^2 - 20k + 16 = 0 \]

Dividing by 4:

\[ k^2 - 5k + 4 = 0 \]

Factoring:

\[ (k - 4)(k - 1) = 0 \]

Thus, \( k = 4 \) or \( k = 1 \).

### Summary of Values of \( k \):
1. From the first equation: \( k = -\frac{10}{9}, 2 \)
2. From the second equation: \( k = 3, 5 \)
3. From the third equation: \( k = -\frac{1}{2}, 1 \)
4. From the fourth equation: \( k = 1, 4 \)